1 Bidirectional User Throughput Maximization Based on ...Light-Fidelity (LiFi) is a subset of OWCs, and it is a bidirectional, high-speed and fully networked wireless communication

1

Bidirectional User Throughput Maximization Basedon Feedback Reduction in LiFi Networks

Mohammad Dehghani Soltani, Xiping Wu, Majid Safari, and Harald Haas

Abstract

Channel adaptive signalling, which is based on feedback, can result in almost any performance

metric enhancement. Unlike the radio frequency (RF) channel, the optical wireless communications

(OWCs) channel is fairly static. This feature enables a potential improvement of the bidirectional user

throughput by reducing the amount of feedback. Light-Fidelity (LiFi) is a subset of OWCs, and it is

a bidirectional, high-speed and fully networked wireless communication technology where visible light

and infrared are used in downlink and uplink respectively. In this paper, two techniques for reducing

the amount of feedback in LiFi cellular networks are proposed, i) Limited-content feedback (LCF)

scheme based on reducing the content of feedback information and ii) Limited-frequency feedback

(LFF) based on the update interval scheme that lets the receiver to transmit feedback information after

some data frames transmission. Furthermore, based on the random waypoint (RWP) mobility model,

the optimum update interval which provides maximum bidirectional user equipment (UE) throughput,

has been derived. Results show that the proposed schemes can achieve better average overall throughput

compared to the benchmark one-bit feedback and full-feedback mechanisms.

I. INTRODUCTION

The ever increasing number of mobile-connected devices, along with monthly global data

traffic which is expected to be 35 exabytes by 2020 [1], motivate both academia and industry to

invest in alternative methods. These include mmWave, massive multiple-input multiple-output

(MIMO), free space optical communication and Light-Fidelity (LiFi) for supporting future growing

data traffic and next-generation high-speed wireless communication systems. Among these technol-

This work has been submitted to IEEE Transactions on Communications. The authors are with the LiFi Research and

Development Center, Institute for Digital Communications, The University of Edinburgh. (E-mail: [email protected];

[email protected]; [email protected]; [email protected]).

July 14, 2018 DRAFT

arX

iv:1

708.

0332

4v1

[cs

.IT

] 1

0 A

ug 2

017

2

ogies, LiFi is a novel bidirectional, high-speed and fully networked wireless communication

technology. LiFi uses visible light as the propagation medium in downlink for the purposes

of illumination and communication. It may use infrared in uplink in order to not affect the

illumination constraint of the room, and also not to cause interference with the visible light in the

downlink [2]. LiFi offers considerable advantages in comparison to radio frequency (RF) systems.

These include the very large, unregulated bandwidth available in the visible light spectrum, high

energy efficiency, and rather straightforward deployment with off-the-shelf light emitting diode

(LED) and photodiode (PD) devices at the transmitter and receiver ends respectively, enhanced

security as the light does not penetrate through opaque objects [3]. These notable benefits of

LiFi have made it favourable for recent and future research.

It is known that utilizing channel adaptive signalling can bring on enhancement in almost any

performance metric. Feedback can realize many kinds of channel adaptive methods that were

considered impractical due to the problem of obtaining instantaneous channel state information

(CSI) at the access point (AP). Studies have proven that permitting the receiver to transmit a

small amount of information or feedback about the channel condition to the AP can provide

near optimal performance [4]–[7]. Feedback conveys the channel condition, e.g., received power,

signal-to-noise-plus-interference ratio (SINR), interference level, channel state, etc., and the AP

can use the information for scheduling and resource allocation. The practical systems using

this strategy, also known as limited-feedback (LF) systems, provide similar performance as the

impractical systems with perfect CSI at the AP.

It is often inefficient and impractical to continuously update the AP with the user equipment

(UE) link condition. However, to support the mobility, it is also essential to consider the

time-varying nature of channels for resource allocation problems to further enhance the spectral

efficiency. With limited capacity, assignment of many resources to get CSI would evacuate

the resources required to transmit actual data, resulting in reduced overall UE throughput [8].

Therefore, it is common for practical wireless systems to update the CSI less frequently, e.g.,

only at the beginning of each frame. Many works have been done to reduce the amount of

DRAFT July 14, 2018

3

feedback in RF, however, very few studies are done to lessen the amount of feedback in optical

wireless channels (OWCs).

A. Literature Review and Motivation

An overview of LF methods in wireless communications has been introduced in [7]. The key

role of LF in single-user and multi-user scenarios for narrowband, wideband communications

with both single and multiple antennas has been discussed in [7]. Two SINR-based limited-feedback

scheduling algorithms for multi-user MIMO-OFDM in heterogeneous network is studied in

[9] where UEs feed back channel quality information in the form of SINR. To reduce the

amount of feedback, nearby UEs grouping and adjacent subcarrier clustering strategies have

been considered. In [10], three limited feedback resource allocation algorithms are evaluated

for heterogeneous wireless networks. These resource allocation algorithms try to maximize the

weighted sum of instantaneous data rates of all UEs over all cells. The authors in [11] proposed

the ordered best-K feedback method to reduce the amount of feedback. In this scheme, only the

K best resources are fed back to the AP.

An optimal strategy to transmit feedback based on outdated channel gain feedbacks and

channel statistics for a single-user scenario has been proposed in [12]. Other approaches are

transmission of the quantized SINR of subcarriers which is the focus of [13] and [14]; and

subcarrier clustering method which is developed in [15] and [16]. In [17], the subcarrier clustering

technique has been applied to the OWCs to reduce the amount of feedback by having each user

send the AP the information of candidate clusters. A simple and more realizable solution, which

is proposed in [18]–[20], is to inform the AP only if their SINR exceed some predetermined

threshold. This is a very simple approach with only a one bit per subcarrier feedback. One-bit

feedback method is very bandwidth efficient. However, using more feedback can provide slight

downlink performance improvement but at the cost of uplink throughput degradation as discussed

in [18]. The benefits of employing only one bit feedback per subcarrier and the minor data rate

enhancements of downlink using more feedback bits are analyzed in [21]. A one-bit feedback

July 14, 2018 DRAFT

4

scheme for downlink OFDMA systems has been proposed in [22]. It specifies whether the

channel gain exceeds a predefined threshold or not. Then, UEs are assigned priority weights,

and the optimal thresholds are chosen to maximize the weighted sum capacity. A problem linked

to one-bit feedback technique is that there is a low probability that none of the UEs will report

their SINR to the AP so that leaving the scheduler with no information about the channel

condition. This issue can be solved at the expense of some extra feedback and overhead by the

multiple-stage version of the threshold-based method proposed in [23].

The limited feedback approaches mentioned above are applicable to LiFi networks. However,

due to fairly static behavior of LiFi channels, the feedback can be reduced further without any

downlink throughput degradation.

B. Contributions and Outcomes

In order to get the maximum bidirectional throughput, the amount of feedback should be

optimized in terms of both quantity and update interval. In this paper, we proposed two methods

to reduce the feedback information. The main contributions of this paper are outlined as follows.

• Proposing the modified carrier sense multiple access with collision avoidance (CSMA/CA)

protocol suitable for uplink of LiFi networks.

• Proposing the limited-content feedback (LCF) scheme for LiFi networks which shows a close

downlink performance to the full-feedback (FF) mechanism and even lower overhead compared

to one-bit feedback technique.

• Proposing the limited-frequency feedback (LFF) scheme based on sum-throughput of uplink

and downlink maximization. Deriving the optimum update interval for random waypoint (RWP)

mobility model and investigating the effects of different parameters on it.

II. SYSTEM MODEL

A. Optical Attocell System Configuration

A bidirectional optical wireless communication has been considered in this study. In the

downlink, visible light is utilized for the purpose of both illumination and communication, while

DRAFT July 14, 2018

5

𝜓𝑖𝑗

Φ1 2⁄

𝑦

𝑥

𝑧

0.5𝑎

0.5𝑎

ℎ

AP𝑖

(0,0,0)

(𝑥𝑗 , 𝑦𝑗 , 0)

−0.5𝑎

⋯

⋯

⋱ ⋱

−0.5𝑎 𝐧rx

(𝑋𝑖 , 𝑌𝑖 , ℎ)

𝐧tx 𝑑𝑖𝑗 𝛼𝑖𝑞

𝛽𝑞𝑗

𝑑𝒜𝑞

𝜓𝑞𝑗

IR LED

PD

𝜑𝑖𝑗

Φ1 2⁄

𝜑

𝜓 𝜑𝑖𝑞

𝐧rx

𝐧tx

Fig. 1: Geometry of light propagation in LiFi networks. Downlink (consist of LOS and NLOS components) and uplink (including

LOS component) are shown with black and red lines, respectively.

in the uplink data are transmitted through infrared light in order not to affect the illumination

constraint of the room. The geometric configuration of the downlink/uplink in an indoor optical

attocell network is shown in Fig. 1. The system comprises multiple LED transmitters (i.e., APs)

arranged on the vertexes of a square lattice over the ceiling of an indoor network and there is

a PD receiver on UE. The LEDs are assumed to be point sources with Lambertian emission

patterns. To avoid nonlinear distortion effects, the LEDs operate within the linear dynamic range

of the current-to-power characteristic curve. In addition, the LEDs are assumed to be oriented

vertically downwards, and the UE are orientated upward to the ceiling. Under this condition, the

channel model for both downlink and uplink is the same. One AP is only selected to serve the

UE based on the UE location. An optical attocell is then defined as the confined area on the UE

plane in which an AP serves the UE. Frequency reuse (FR) plan is considered in both downlink

and uplink to reduce the co-channel interference and also guarantee the cell edge users data rate.

Further details about FR plan can be found in [24] and [25].

Power- and frequency-based soft handover methods for visible light communication networks

are proposed to reduce data rate fluctuations as the UE moves from one cell to another [26]. We

consider power-based soft handover with the decision metric introduced in [27] as |γı−γi| < α,

July 14, 2018 DRAFT

6

where γı and γi are the SINR of the serving AP and adjacent APs, respectively; and α is

the handover threshold. As a results the cell boundaries shape a circle with the radius of rc.

According to the considered soft handover scheme, when the difference of SINR from two APs

goes below the threshold, handover will occur.

The received optical signal at the PD consists of line of sight (LOS) and/or non-line of sight

(NLOS) components. The LOS is a condition where the optical signal travels over the air directly

from the transmitter to the UE, while the NLOS is a condition where the optical signal is received

at the UE just by means of the reflectors. These two components are characterized as follows.

B. Light Propagation Model

The direct current (DC) gain of the LOS optical channel between the ith LED and the jth

PD is given by:

HLOS,i,j =

(m+ 1)A

2πd2ij

cosmφijgfg(ψij) cosψij, 0 ≤ ψij ≤ Ψc

0, ψij > Ψc

, (1)

where A, dij , φij and ψij are the physical area of the detector, the distance between the ith

transmitter and the jth receiver surface, the angle of radiance with respect to the axis normal to

the ith transmitter surface, and the angle of incidence with respect to the axis normal to the jth

receiver surface, respectively. In (1), gf is the gain of the optical filter, and Ψc is the receiver

field of view (FOV). In (1), g(ψi) = ς2/ sin2 Ψc for 0 ≤ ψi ≤ Ψc, and 0 for ψi > Ψc, is the

optical concentrator gain where ς is the refractive index; and also m = −1/ log2(cos Φ1/2) is

the Lambertian order where Φ1/2 is the half-intensity angle [28]. The radiance angle φij and the

incidence angle ψij of the ith LED and jth UE are calculated using the rules from analytical

geometry as cosφij = dij · ntx/‖dij‖ and cosψij = −dij · nrx/‖dij‖, where ntx = [0, 0,−1]

and nrx = [0, 0, 1] are the normal vectors at the transmitter and jth receiver planes, respectively

and dij denotes the distance vector between ith LED and the jth UE and · and ‖ · ‖ denote the

inner product and the Euclidean norm operators, respectively.

DRAFT July 14, 2018

7

In NLOS optical links, the transmitted signal arrives at the PD through multiple reflections.

In practice, these reflections contain both specular and diffusive components. In order to keep a

moderate level of analysis, first-order reflections only are considered in this study. A first-order

reflection consists of two segments: i) from the LED to a small area dAq on the wall; and ii)

from the small area dAq to the PD. The DC channel gain of the first-order reflections is given

by:

HNLOS,i,j =

∫Aq

ρq(m+1)A

2π2d2iqd

2qj

cosmφiq cosψqjgfg(ψqj) cosαiq cos βqjdAq, (2)

where Aq denotes the total walls reflective area; ρq is the reflection coefficient of the qth reflection

element; diq is the distance between the ith LED and the qth reflection element; dqj is the distance

between the qth reflection element and the jth UE; φiq and ψiq are the angle of radiance and

the angle of incidence between the ith LED and the qth reflective element, respectively; and φqj

and ψqj are the angle of radiance and the angle of incidence between the qth reflective element

and the jth UE, respectively [29]. The channel gain between APi and UEj is comprised of both

LOS and NLOS components that is expressed as:

Hi,j = HLOS,i,j +HNLOS,i,j. (3)

Note that due to symmetry of downlink and uplink channels, (1)-(3) are valid for both downlink

and uplink.

C. Low Pass Characteristic of LED

The frequency response of an off-the-shelf LED is not flat and is modeled as a first order low

pass filter as, HLED(w) = e−w/w0 , where w0 is the fitted coefficient [30]. The higher the value of

w0, the wider the 3-dB bandwidth, B3dB. The 3-dB bandwidth of typical LEDs is low, however,

the modulation bandwidth, B, can be multiple times greater than B3dB thanks to utilization of

OFDM. In this paper, we consider OFDMA for two purposes: i) to alleviate the low pass effect

of LED and ii) to support multiple access. The frequency response of LED on kth subcarrier

can be obtained as:HLED,k = e−2πkBd,n/Kw0 , (4)

where K is the total number of subcarriers and Bd,n is the downlink bandwidth of nth FR plan.

July 14, 2018 DRAFT

8

𝑟c

Φ1 2⁄

𝑣𝑡

AP

ℎ

Polar axis

Cell boundary

UE

𝑟0

𝑑0

𝑟(𝑡)

𝑃1

𝑃0

(a)

𝑟c

Φ1 2⁄

𝑣𝑡

AP

ℎ

Polar axis

Cell boundary

UE

𝑟0

𝑑0

𝑟(𝑡)

𝑃1

𝑃0

𝑃1 𝑃0

𝑟0 𝑟(𝑡)

Cell boundary

AP

UE 𝑣𝑡

𝑟c

𝜃

(b)

Fig. 2: RWP movement model.

D. Receiver Mobility Model

We considered the RWP model which is a commonly used mobility model for simulations of

wireless communication networks [31]. The RWP mobility model is shown in Fig. 2. According

to the RWP model, the UE’s movement from one waypoint to another waypoint complies with

a number of rules, including i) the random destinations or waypoints are chosen uniformly with

probability 1/(πr2c); ii) the movement path is a straight line; and iii) the speed is constant during

the movement. The RWP mobility model can be mathematically expressed as an infinite sequence

of triples: {(P`−1,P`, v`)}`∈N where ` denotes the `th movement period during which the UE

moves between the current waypoint P`−1 =(x`−1, y`−1, 0) and the next waypoint P` = (x`, y`, 0)

with the constant velocity V` = v. RWP model is more realistic scenario and has been used in

many studies for modeling the mobility of UE [32], [33].

The UE distance at time instance t from the AP is d(t) = (r2(t) + h2)1/2, where r(t) =

(r20 + v2t2 − 2r0vt cos θ)1/2 with θ=π− cos−1

(~r0·~v| ~r0||~v|

); ~r0 is the initial UE distance vector from

the cell center at t=0 with |~r0| = r0; and ~v is the vector of UE’s velocity with |~v| = v. Here,

r0 has the probability distribution function (PDF) of fR0(r0)=2r0/r2c and θ is chosen randomly

from a uniform distribution with PDF of fΘ(θ)=1/π. For notation simplicity, the dependency

of the equations to time is omitted unless it is confusing.

DRAFT July 14, 2018

9

III. DOWNLINK THROUGHPUT CALCULATION

The channel access protocol in the downlink is assumed to be orthogonal frequency division

multiple access (OFDMA) based on DCO-OFDM so as to support downlink multiple access

simultaneously. The modulated data symbols of different UEs, Xk, are arranged on K subcarriers

of the OFDMA frame, X . Then, the inverse fast Fourier transform (IFFT) is applied to the

OFDMA frame to obtain the time domain signal x. For optical systems that perform intensity

modulation, the modulated signal, x, must be both real and positive [34]. This requires two

constraints on the entities of OFDMA frame: i) X(0) = X(K/2) = 0, and ii) the Hermitian

symmetry constraint, i.e., X(k) = X∗(K − k), for k 6= 0, where (·)∗ denotes the complex

conjugate operator. Therefore, the OFDMA frame is X = ζ[0, X1, ..., XK/2−1, 0, X∗K/2−1, ..., X

∗1 ],

the normalizing factor, ζ =√K/(K − 2), is multiplied since the 0th and (K/2)th samples require

no energy. Note that the number of modulated subcarriers bearing information is K/2 − 1.

Afterwards, a moderate bias relative to the standard deviation of the AC signal x is used as

xDC = η√E[x2] [35]. The signal x = xDC + x is then used as the input of an optical modulator.

Let Hj = [Hi,j], for i = 1, 2, ..., NAP, be the downlink visible light channel gain vector from

all APs to the UEj . The UEj is connected to APı based on the maximum channel gain criterion

so that ı = argi max(Hj). Afterwards, the embedded scheduler algorithm in APı allocates a

number of subcarriers to the UEj based on its requested data rate and its link quality. In this

study, a fair scheduling method for OFDMA-based wireless systems is considered [36], [37].

The scheduler assigns the kth resource to jth UE according to the following metric:

j = arg maxi

Rreq,j

Ri

, (5)

where Ri is the average data rate of ith UE before allocating the kth resource, and Rreq,j is the

request data rate of UEj .

Throughout this study, we consider LiFi systems transmitting data based on DC-biased optical

OFDM, for which the upper bound on the achievable data rate can be expressed in a Shannon

capacity expression form as a function of electrical SINR as shown in [38]. Assume the effect

July 14, 2018 DRAFT

10

of clipping noise is negligible, the downlink rate of UEj after scheduling can be obtained as:

Rd,j =Bd,n

K

K/2−1∑k=1

log2 (1 + sj,kγd,j,k) , (6)

where sj,k =1 if kth subcarrier is allocated to the UEj otherwise sj,k =0; γd,j,k is the SINR of

UEj on kth subcarrier serving by APı. In communication systems, SINR is defined as the ratio

of the desired electrical signal power to the total noise and interference power and is an important

metric to evaluate the connection quality and the transmission data rate. Denoting Pelec,ı,j,k as

the received electrical power of jth UE on kth subcarrier, then, γd,j,k = Pelec,ı,j,k/(σ2j,k +Pint,j),

where σ2j,k=N0Bd,n/K, is the noise on kth subcarrier of UEj , and N0 is the noise power spectral

density; Pint,j is the interference from other APs on jth UE. It is assumed that the APs emit the

same average optical power and the total transmitted electrical power is equally allocated among

K − 2 subcarriers so that the received electrical power on kth subcarrier of jth UE is equal to

Pelec,ı,j,k = R2PDP

2d,optH

2ı,j,kH

2LED,k/(η

2(K − 2)), where Pd,opt is the transmitted optical power;

RPD and η are the PD responsivity and conversion factor, respectively; Hı,j,k is the frequency

response of channel gain on kth subcarrier. It includes both LOS and the first order reflections.

Accordingly, the received SINR of jth UE on kth subcarrier can be expressed as:

γd,j,k=R2

PDP2d,optH

2ı,j,kH

2LED,k

(K−2)η2σ2j,k+

∑i∈SAP,ı

R2PDP

2d,optH

2i,j,kH

2LED,k

. (7)

where SAP,ı is the set of all other APs using the same frequencies as the APı.

IV. UPLINK THROUGHPUT CALCULATION

A. Uplink Access Protocol

In this study, CSMA/CA is considered as the uplink access protocol. CSMA/CA is a multiple

access protocol with a binary slotted exponential backoff strategy being used in wireless local area

networks (WLANs) [39]. This is known as the collision avoidance mechanism of the protocol.

In CSMA/CA, a UE will access the channel when it has data to transmit. Thus, this access

protocol uses the available resources efficiently. Once the UE is allowed to access the channel,

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11

it can use the whole bandwidth. However, this access protocol cannot directly be used in LiFi

networks, because it results in severe “hidden node” problem. Here, we applied two simple

modifications to CSMA/CA to minimize the number of collisions in LiFi networks. Firstly,

the request-to-send/clear-to-send (RTS/CTS) packet transmission scheme, which is optional in

WLANs should be mandatory in LiFi networks. This is the only way that UEs can notice

that the channel is busy in LiFi networks. The reason behind this is that different wavelengths

are employed in downlink and uplink of LiFi networks, visible light and infrared, respectively.

Thus, the PD at the UE is tuned for visible light and cannot sense the channel when another UE

transmits via infrared. Secondly, the AP transmits a channel busy (CB) tone to inform the other

UEs that the channel is busy. In the following, the modified CSMA/CA is described in detail.

B. Brief Description of the Access Protocol

In CSMA/CA, UEs listen to the channel prior to transmission for an interval called distributed

inter-frame space (DIFS). Then, if the channel is found to be idle, the UEs generate a random

backoff, Bj , for j = 1, 2, . . . , N , where N is the number of competing UEs. The value of Bj is

uniformly chosen in the range [0, w−1], where w is the contention window size. Let B = [Bj]1×N ,

be the backoff vector of the UEs. After sensing the channel for time interval DIFS, UEj should

wait for Bj × tslot seconds, where tslot is the duration of each time slot. Obviously, the UE

with lowest backoff is prior to transmit, i.e., u1th UE, where u1 = argj min(B). Then, u1th

UE sends the RTS frame to the AP before N − 1 other UEs. If the RTS frame received at the

AP successfully, it replies after a short inter-frame space (SIFS) with the CTS frame. The u1th

UE only proceeds to transmit the data frame, after the time interval of SIFS, if it receives the

CTS frame. Eventually, an ACK is transmitted after the period of SIFS by the AP to notify the

successful packet reception. The AP transmits the CB tone simultaneously with the reception of

RTS packet. The UEs that can hear the CB tone will freeze their backoff counter. The backoff

counter will reactivated when the channel is sensed idle again after the period of DIFS. If the

AP does not transmit the CB tone, the u2th UE who cannot hear the u1th UE, will start to send

July 14, 2018 DRAFT

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RTS frame after waiting for Bu2 × tslot seconds. Here, u2th UE is called the hidden UE and a

collision occurs if (Bu2 − Bu1) × tslot < tRTS, where tRTS is the RTS frame transmission time

which is directly proportional to the length of RTS frame, LRTS, and inversely proportional to

the uplink rate.

C. Uplink Throughput

In the modified CSMA/CA for LiFi networks, collision only occurs if the backoff time of

at least two UEs reach to zero simultaneously. Thus, they transmit at the same time and the

packets collide. The analysis of normalized throughput and collision probability is the same as

the analysis provided in [40]. In the following, we only provide a summary of the equations and

further detail is provided in [40]. The normalized uplink throughput is given as:

Tu =PtPsE[tD]

(1− Pt)tslot + PtPsE[ts] + Pt(1− Ps)tc, (8)

where Pt = 1− (1− τ)N is the probability of at least one transmission in the considered backoff

slot time, Ps = Nτ(1− τ)N−1/Pt is the probability of successful transmission, and τ = 2w+1

is the probability that a UE transmits on a randomly chosen slot time. In (8), E[tD],E[ts] and

tc are the average transmission time of data packet, average successful transmission time and

collision time, respectively. Assuming that all data packets have the same length, then:

E[ts] = ts = tRTS + SIFS + tdely + tCTS + SIFS + tdely + tHDR + tD + SIFS

+ tdely + tACK + DIFS + tdely, E[tD] = tD, tc = tRTS + DIFS + tdely

(9)

where tdely is the propagation delay. Note that the packet header includes both physical and

MAC header. Finally, the uplink throughput of jth UE can be obtained as follows:

Ru,j =TuBu,n

Nlog2 (1 + γu,j) . (10)

where Bu,n is the uplink bandwidth of nth FR plan and γu,j is the SINR at the AP when

communicating with UEj and it is given as:

γu,j =(RPDPu,optHı,j)

2

η2N0Bu,n +∑

j∈Π (RPDPu,optHi,j)2 , (11)

DRAFT July 14, 2018

13

where Π is the set of other UEs using the same bandwidth as UEj and communicating with

ith AP, (i 6= ı), simultaneously with UEj; and Pu,opt is the transmitted uplink power which is

assumed to be the same for all UEs.

V. FEEDBACK MECHANISM

Over the last few years, studies have repeatedly illustrated that permitting the receiver to

send some information bits about the channel conditions to the transmitter can allow effective

resource allocation and downlink throughput enhancement. This feedback information is usually

the SINR of a subcarrier at the receiver [7], [10]. However, sending this information is in cost

of uplink throughput degradation. Therefore, there is a trade-off between downlink and uplink

throughput when the amount of feedback varies. Let’s define the feedback ratio, ε, as the ratio

of total feedback time and total transmission time as:

ε =

∑tfb

ttot

, (12)

where tfb is the feedback duration. where tfb is the feedback duration. Fig. 3-(a) denotes a general

feedback mechanism, in which feedback information is transmitted periodically after an interval

of tu. Denoting that the denominator of (12) is the total transmission time which is equal to

ttot = (ND +Nf)tfr, where ND and Nf are the number of data and feedback frames in the total

transmission time. The total feedback time is Σtfb = Nftfb. Replacing these equations in (12),

the feedback ratio can be obtained as:

ε =Nftfb

(ND +Nf)tfr=

tfb(1 + ND

Nf

)tfr. (13)

Since ttot = (ND + Nf)tfr = Nftu, then 1 +ND

Nf

=tutfr

, and substituting it in (13), it can be

simplified as:ε =

tfbtu. (14)

Then, the uplink throughput of UEj in consideration of feedback is given by:

Ru,j =

(1− tfb

tu

)TuBu,n

Nlog2 (1 + γu,j) . (15)

July 14, 2018 DRAFT

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Data Feedback … Feedback

Frame #1 Frame #2 Frame #𝑀

Data

Frame #𝑀 + 1

𝑡fb

Data Feedback Data Feedback

Data Feedback

𝐾

2 -1 bits feedback

(b) Full feedback scheme

Data Data … Data Data

(c) One-bit feedback scheme

Data … Data Data

One Byte feedback

(d) Proposed limited-content feedback (LCF) scheme

Data

Frame #1 Frame #2 Frame #𝑀 Frame #𝑀 + 1

𝑡fb 𝑡u

𝑡fr

Data Feedback Data … Data Feedback

𝑡fr

(a) Feedback mechanism

Fig. 3: Feedback schemes.

Due to the use of DCO-OFDM modulation, the AP requires the SINR information of K/2−1

subcarriers. The extreme and least cases for sending the SINR information are full feedback

(FF) and one-bit fixed-rate feedback, respectively. These schemes are shown in Fig. 3-(b) and

Fig. 3-(c). In the FF scheme, UEs send the SINR of all subcarriers at the beginning of each

data frame. Obviously, this impractical method produces huge amount of feedback. According

to one-bit feedback technique, the AP sets a threshold for all UEs. Each UE compares the value

of its SINR to this threshold. When the SINR exceeds the threshold, a ‘1’ will be transmitted to

the AP; otherwise a ‘0’ will be sent. The AP receives feedback from all UEs and then randomly

selects a UE whose feedback bit was ‘1’. If all the feedback bits received by the AP are zero,

then no signal is transmitted in the next time interval. However, in this case the AP can also

randomly chooses a UE for data transmission, although for large number of UEs this method

has vanishing benefit over no data transmission when all the received feedback bits are ‘0’ [41].

As can be induced from (14), the feedback ratio can generally be reduced by means of

either decreasing the content of feedback or increasing the update interval. In the following, we

propose the limited-content feedback (LCF) and limited-frequency feedback (LFF) techniques.

The former is based on reducing the feedback information in each frame and the latter is based

on increasing the update interval.

DRAFT July 14, 2018

15

A. Proposed Limited-content feedback (LCF) Scheme

Unlike RF wireless and optical diffused channels, the frequency selectivity of the channel

in LiFi attocell networks is mostly characterized by the limitations of the receiver/transmitter

devices (i.e., PD and LED), rather than the multipath nature of the channel [28]. In order to assess

the frequency response of the free-space optical channels, computer simulations are conducted.

The simulations are performed for a network size of 10 × 10 × 2.15 m3. The network area is

divided equally into nine quadrants with one AP located at the center of each. Assume the center

of xy-plane is located in the center of the room as shown in Fig. 1. The other parameters are

listed in Table I. The normalized frequency response of the channel gain, |Hi,j(f)|2|HLOS,i,j(f)|2 , for a UE

placed at different positions of the room is depicted in Fig. 4. As can be seen, the normalized

frequency response fluctuates around the LOS component and the variation of the fluctuation is

less than 1 dB. Moreover, the channel gain variation is less significant for UEs that are further

away from the walls of the room, due to the lower significance of the first order reflection

[25]. Accordingly, the frequency selectivity of LiFi channels is mainly confined by LED and PD

components, and the frequency selectivity of these devices are fairly static. The average received

power at the UE is much more dynamic and is significantly dependent on the position of the

UE. Therefore, by only updating the average power, a reasonable estimate of the SINR of all

the subcarriers can be obtained. This idea forms the foundation of our LCF scheme.

Fig. 3-(d) represents the principal working mechanism of our proposed LCF scheme. According

to the LCF scheme, when a UE connects to an AP, it sends the SINR information of all subcarriers

only once at the beginning of the first frame. For the following frames and as long as the UE

is connected to the same AP, it only updates the scheduler on its received average power (i.e.,

the DC channel component). Once the UE connects to a new AP, it will transmit the SINR

information of all subcarriers again. The proposed LCF scheme then simply scales the individual

SINR values received in the next frames such that the total average power matches the updated

average power [42]. Thus, the estimated SINR on kth subcarrier of jth UE at time instance t is

July 14, 2018 DRAFT

16

TABLE I: Simulation Parameters

Parameter Symbol Value

Network space – 10× 10× 2.15 m3

Number of APs NAP 9

Cell radius rc 2.35 m

LED half-intensity angle Φ1/2 60◦

Receiver FOV Ψc 90◦

Physical area of a PD A 1 cm2

Gain of optical filter gf 1

Refractive index ς 1

PD responsivity RPD 1 A/W

Reflection coefficient ρq 0.85

Number of subcarriers K 2048

Transmitted optical power Pd,opt 8 W

Downlink FR bandwidth Bd,n 10 MHz

Fitted coefficient w0 45.3 Mrad/s

Conversion factor η 3

Noise power spectral density N0 10−21 A2/Hz

given as:

γd,j,k(t) ≈ γd,j,k(0)× γd,j,0(t)

γd,j,0(0), (16)

where γd,j,k(0) is the downlink SINR of jth UE on kth subcarrier at t = 0. The scheduler uses

this estimated SINR information for subcarrier allocation according to (5).

The most salient difference between the LCF technique and one-bit feedback method is that

the AP does not have any knowledge about the SINR value of each subcarrier and it just knows

that the SINR is above or lower than a predetermined threshold for one-bit feedback technique.

However, thanks to the use of LCF approach, the AP can have an estimation of the SINR value

for each subcarrier. In order to compare the downlink performance of FF, one-bit feedback and

LCF, Monte-Carlo simulations are executed. The simulation tests are carried out 103 times per

various number of UEs, and with the parameters given in Table I. In each run, the UEs’ locations

are chosen uniformly random in the room. Once they settle in the new locations, they update

DRAFT July 14, 2018

17

Frequency, f [Hz]105 106 107 108N

orm

aliz

ed c

hann

el g

ain

[dB

]

-0.5

0

0.5

1 x=0, y=0 (Room center)x=1, y=1x=2,y=2x=3,y=3x=4,y=4 (Near wall)

Fig. 4: Normalized channel gain, |Hı,j(f)|2

|HLOS,ı,j(f)|2, for different room positions.

Total number of UEs5 20 40 60 80 100

Ave

rage

dow

nlin

k th

roug

hput

, [M

bps]

15

20

25

30

35

40

Full feedback schemeProposed LCF schemeOne-bit feedback scheme

Rreq

=20 Mbps

Rreq

=40 Mbps

Fig. 5: Average downlink throughput for different feedback schemes (average request data rate: 20 Mbps and 40 Mbps).

the AP about their subcarrier SINR as explained. Then, the AP, reschedule the resources based

on (5). The request data rate of UEs are assumed to be the same. Fig. 5 illustrates the average

downlink throughput versus different number of UEs for LCF, FF and on-bit feedback schemes.

As can be seen from the results, the performance of the LCF is better than the one-bit feedback

scheme and nearly similar to FF scheme. As the number of UEs increase the gap between

the considered feedback schemes also increases. However, the LCF follows the FF fairly good

especially for low data request rate. Moreover, compared to the one-bit feedback technique, the

LCF scheme occupies less portion of the uplink bandwidth.

B. Proposed Limited-frequency feedback (LFF) Scheme

Due to fairly static feature of LiFi channels, the UE can update the AP about its channel

condition less frequently, especially when the UE is immobile or it moves slowly [43]. Based on

July 14, 2018 DRAFT

18

the information of UE’s velocity, we aim to find the appropriate channel update interval, tu, so that

the expected weighted average sum throughput of uplink and downlink per user is maximized.

Weighted sum throughput maximization is commonly used to optimize the overall throughput

for bidirectional communications [44], [45]. The optimization problem (OP) is formulated as:

maxtu

(E[r0],[θ]

[1

N

N∑j=1

(wdRd,j(tu) + wuRu,j(tu)

)]), (17)

where Rd,j and Ru,j are the average downlink and uplink throughput of jth UE, respectively; Note

that [r0] = [r01, · · · , r0N ] and [θ] = [θ1, · · · , θN ] are random variable vectors with i.i.d entities;

E[r0],[θ][·] is the expectation with respect to the joint PDF f([r0], [θ])=f(r01, · · · , r0N , θ1, · · · , θN).

Since r0j’s and θj’s are i.i.d, we have f([r0], [θ]) = fR0(r0j)fΘ(θj)∏

i 6=j fR0(r0i)fΘ(θi), where

fR0(r0j) and fΘ(θj) are described in Section II. The expectation can go inside the summation,

then, we have E[r0],[θ]

[Rd,j(tu)

]= Er0j ,θj

[Rd,j(tu)

]for downlink and E[r0],[θ]

[Ru,j(tu)

]=

Er0j ,θj[Ru,j(tu)

]for uplink. Since r0j’s and θj’s are i.i.d, then:

Er01,θ1[Rd,1(tu)

]= · · ·=Er0N ,θN

[Rd,N(tu)

],Er0,θ

[Rd(tu)

]Er01,θ1

[Ru,1(tu)

]= · · ·=Er0N ,θN

[Ru,N(tu)

],Er0,θ

[Ru(tu)

].

After substituting above equations in (17) and some manipulations, the OP can be expressed as:

maxtu

(T = wuEr0,θ

[Ru(tu)

]+ wdEr0,θ

[Rd(tu)

]), (18)

which is not dependent on any specific UEs. The average is calculated over one update interval,

since it is assumed the UE feeds back its velocity information to the AP after each update

interval. The opposite behaviour of Ru and Rd with respect to the update interval (the former

directly and the latter inversely are proportional to the update interval), results in an optimum

point for T . In the following, Ru and Rd are calculated with some simplifying assumptions.

The exact and general state of SINR at the receiver is provided in (7). However, for ease of

analytical derivations, it can be simplified under some reasonable assumptions including: i) the

interference from other APs can be neglected due to employing FR plan, ii) Hi,j,k ≈ HLOS,i,j . The

latter assumption is based on the fact that in OWC systems, HLOS,i,j>>HNLOS,i,j . It was shown

DRAFT July 14, 2018

19

in Fig. 4 that the variation of the frequency response fluctuation around the LOS component

is less than 1 dB. Using Fig. 1, cosφij = cosψij = h/dij , can be substituted in (1), then, the

DC gain of the LOS channel is HLOS,i,j = G0/dm+3ij , where G0 = (m+1)A

2π sin2 Ψcgfς

2hm+1. Hence, the

approximate and concise equation of SINR at kth subcarrier of jth UE is given by:

γj,k ≈Ge

−4πkBd,nKw0(

r2j + h2

)m+3 , (19)

where G =KG2

0R2PDP

2d,opt

(K−2)η2N0Bd,nand rj is the distance between the UEj and the center of the cell which

is located in it. Substituting (19) in (6), the downlink throughput is given as:

Rd,j =Bd,n

K

K2−1∑

k=1

log2

1 + sj,kGe

−4πkBd,nKw0(

r2j + h2

)m+3

. (20)

Noting that typically in LiFi cellular networks using FR, SINR values are high enough, we have:

Rd,j =Bd,n

K

K2−1∑

k=1

sj,k log2

Ge−4πkBd,nKw0(

r2j + h2

)m+3

. (21)

Same approximations can be also considered for uplink throughput. Define Gu = (G0RPDPu,opt)2

η2N0Bu,n,

then, the SINR at the AP is γu,j =Gu/(r2j + h2)m+3. Substituting it in (15), the uplink throughput

of UEj can approximately be obtained as:

Ru,j∼=(

1− tfbtu

)TuBu,n

Nlog2

(Gu(

r2j +h2

)m+3

). (22)

Without loss of generality and for ease of notations, we consider one of the N UEs for the

rest of derivations and remove the subscript j. The average uplink throughput over one update

interval is given as:

Ru =

(1− tfb

tu

)TuBu,n

N

1

tu

∫ tu

0

log2

(Gu

(r2(t) + h2)m+3

)dt

=2(m+ 3)TuBu,n

N

(1− tfb

tu

)(1

2(m+ 3)log2

(Gu

(r2(tu) + h2)m+3

)+r0 cos θ

2vtulog2

(r2(tu)+h2

r20 + h2

)+

1

ln(2)− (h2+ r2

0 sin2 θ)12

vtu ln(2)tan−1

(vtu − r0 cos θ

(h2+ r20 sin2 θ)

12

)− (h2+r2

0 sin2θ)12

vtu ln(2)tan−1

(r0 cos θ

(h2+r20 sin2θ)

12

)).

(23)

The average downlink throughput over one update interval can be obtained as:

July 14, 2018 DRAFT

20

Rd =Bd,n

Ktu

∫ tu

0

kreq∑k=1

log2

Ge−4πkBd,nKw0

(r2(t) + h2)m+3

dt =kreqBd,n

Ktu

∫ tu

0

log2

(Ge−2π(kreq+1)Bd,n/Kw0

(r2(t) + h2)m+3

)dt

=2(m+ 3)kreqBd,n

K

(1

2(m+ 3)log2

(Ge−2π(kreq+1)Bd,n/Kw0

(r2(tu) + h2)m+3

)+r0 cos θ

2vtulog2

(r2(tu)+h2

r20 + h2

)+

1

ln(2)

−(h2 + r20 sin2 θ)

12

vtu ln(2)tan−1

(vtu − r0 cos θ

(h2 + r20 sin2 θ)

12

)− (h2+r2

0 sin2θ)12

vtu ln(2)tan−1

(r0 cos θ

(h2+r20 sin2θ)

12

)).

(24)

where kreq is the required number of subcarriers to be allocated to the UE at t = 0. With the

initial and random distance of r0 from the cell center, the required number of subcarriers can

approximately be obtained as:

kreq∼=

KRreq

Bd,n log2 (G/(r20 + h2)m+3)

(25)

The exact value and proof are given in Appendix-A. Both the average uplink and downlink

throughput given in (23) and (24), respectively, are continuous and derivative in the range

(0, 2rc/v). Therefore, we can express the following proposition to find the optimal update interval

that results in the maximum sum-throughput.

Proposition. Let tu be continuous in the range of (0, 2rc/v). The optimal solution to the OP

given in (18) can be obtained by solving the following equation:

Er0,θ[∂T∂tu

]= wuEr0,θ

[∂T u

∂tu

]+ wdEr0,θ

[∂T d

∂tu

]= 0. (26)

For vtu � h, the root of (26) can be well approximated as:

tu,opt∼=

( 3ln(2)2(m+3)

wutfbTuBu,nC1

wdv2NRreq + C2wuv2TuBu,n

)13

, (27)

where

C1 =

Er0[log2

(Gu

(r20 + h2)m+3

)]Er0[log2

(G

(r20 + h2)m+3

)]Er0,θ

[(h2 + r2

0 sin2 θ)2

(h2 + r20)3

] , C2 = Er0[log2

(G

(r20 + h2)m+3

)].

(28)

Proof: See Appendix-B

As it can be seen from (27), the optimum update interval depends on both physical and MAC

layer parameters. Among them, the UE velocity affects the update interval more than others.

DRAFT July 14, 2018

21

TABLE II: Uplink simulation parameters

Parameter Symbol Value

Transmitted uplink optical power Pu,opt 0.2 W

Uplink FR bandwidth Bu,n 5 MHz

Average length of uplink payload LD 2000 B

Physical header HPHY 128 b

MAC header HMAC 272 b

RTS packet size LRTS 288 b

CTS packet size LCTS 240 b

ACK packet size LACK 240 b

SIFS −− 16 µs

DIFS −− 32 µs

Backoff slot duration tslot 8 µs

Propagation delay tdelay 1 µs

Feedback time tfb 0.8 ms

Let’s fix the other parameters, then tu,opt = Cconst/v23 , where Cconst =

(3ln(2)

2(m+3)wutfbTuBu,nC1

wdRreq+C2wuTuBu,n

)13

. We

study the effect of UE’s velocity and transmitted downlink optical power on the update interval

as illustrated in Fig. 6. Analytical and Monte-Carlo simulations are presented for wu = wd,

N = 5 and with the downlink and uplink simulation parameters given in Table I and Table II,

respectively. For a fixed tu, Monte-Carlo simulations are accomplished 104 times, where in each

run, the UE’s initial position and direction of movement are randomly chosen. Then, for the

considered tu, the expected sum-throughput, T , can be obtained by averaging out over 104 runs.

Afterwards, based on the greedy search and for different tu, varying in the range 0<tu<2rc/v,

Monte-Carlo simulations are repeated. The optimal update interval corresponds to the maximum

sum-throughput. The effect of UE’s velocity on optimal update interval for Rreq = 5 Mbps and

Rreq = 20 Mbps is shown in Fig. 6-(a). Here, we can see the optimal update interval decrease

rapidly as UE’s speed increases, according to v−2/3. Further, Monte-Carlo simulations confirm

the accuracy of analytical results provided in (27). Fig. 6-(b) illustrates the saturated effect of

transmitted optical power on tu,opt. As can be observed, the variation of optimal update interval

July 14, 2018 DRAFT

22

Velocity, [m/s]0.5 1 1.5 2

Opt

imal

upd

ate

inte

rval

, [s]

0.05

0.1

0.15

0.2

0.25

0.3 Analytical results Rreq

=5 Mbps

Monte-Carlo simulation results Rreq

=5 Mbps

Analytical results Rreq

=20 Mbps

Monte-Carlo simulation results Rreq

=20 Mbps

(a) The effect of UE’s velocity on optimal update interval

(Pd,opt = 8watt).

Transmitted downlink optical power of AP, [watt]1 2 3 4 5 6 7 8 9 10

Opt

imal

upd

ate

inte

rval

, [s]

0.08

0.1

0.12

0.14

0.16

(b) The effect of transmitted downlink optical power on

optimal update interval (v = 1 m/s).

Fig. 6: The effects of UE’s velocity and downlink optical power on optimal update interval for Rreq = 5 Mbps and Rreq = 20

Mbps, and N=5.

6

0 0.25 0.5 0.75 1

Opt

imal

upd

ate

inte

rval

, [s]

0.09

0.12

0.15

0.18

0.21

0.24

Analytical results, v=0.7 m/sMonte-Carlo simulation results, v=0.7 m/sAnalytical results, v=1 m/sMonte-Carlo simulation results, v=1 m/sAnalytical results, v=1.4 m/sMonte-Carlo simulation results, v=1.4 m/s

Fig. 7: Optimal update interval versus overload parameter, λ, for different UE’s velocity (N = 5).

due to alteration of Pd,opt is less than 30 ms. From both Fig. 6-(a) and Fig. 6-(b), it can be

deduced the lower Rreq, the higher tu,opt.

Now let’s consider an overloaded multi-user scenario with N users. The fair scheduler introduced

in (5) tries to equalize the rate of all UEs. For high number of subcarriers, the UEs achieve

approximately the same data rate. Accordingly, the on average achieved data rate of UEs in an

overloaded network for high number of subcarriers would nearly be λRreq, where 0 < λ < 1.

This system is equivalent to a non-overloaded multi-user system where all UEs have achieved

on average their request rate of λRreq. Then, the approximate optimal update interval that results

in near-maximum sum-throughput is given as:

tu,opt∼=

( 3ln(2)2(m+3)

wutfbTuBu,nC1

wdv2NλRreq + C2wuv2TuBu,n

)13

. (29)

Analytical and Monte-Carlo simulations of an overloaded system are shown in Fig. 7. Three

DRAFT July 14, 2018

23

Velocity, [m/s]0 0.5 1 1.5 2

Exp

ecte

d ov

eral

l thr

ough

put,

[Mbp

s]

17

18

19

20

21

22

23

24

Scenario I: Without update intervalScenario II: Fixed update interval, 10 msScenario III: LFF with optimum update interval

Fig. 8: Expected overall throughput versus UE’s velocity for three scenarios; and Rreq = 20 Mbps, wu = wd = 1.

speed values are chosen around the average human walking speed which is 1.4 m/s [46]. Note

that to obtain an overloaded system either the number of UEs or their request data rate can be

increased. In the results shown in Fig. 7, we fixed the number of UEs to N = 5 and increase

their Rreq. As can be inferred from these results, as the network becomes more overloaded, the

optimal update interval should be increased. The reason is that in an overloaded network, due

to lack of enough resources updating the AP frequently is useless and it just wastes the uplink

resources.

To verify the significance of update interval in practical systems, three scenarios have been

considered. Scenario I: a system without any update interval; Scenario II: a system with the

conventional fixed update interval but without looking at the UE’s velocity; Scenario III: a system

with the proposed update interval and adjustable with the UE’s velocity. For these scenarios,

Monte-Carlo simulation results of expected sum-throughput versus different UE’s velocity have

been obtained and presented in Fig. 8. In scenario I, the UEs only update the AP once at

the start of the connection by transmission of the SINR information of K/2 − 1 subcarriers.

For scenario II, the fixed update interval is considered to be tu = 10 ms and independent of

UE’s velocity. Fixed update interval is currently used in LTE with tu = 10 ms by transmission

of one-bit feedback information at the beginning of every frame [47]. It is worth mentioning

that for practical wireless systems, it is common to transmit feedback frequently, e.g., at the

July 14, 2018 DRAFT

24

beginning of each frame regardless of the UE channel variation and velocity. As can be seen

from the results, the proposed LFF scheme outperforms the conventional method with fixed

update interval. For low speeds (up to 0.5 m/s), the conventional fixed update interval even falls

behind the system without any update interval. This is due to redundant feedback information

being sent to the AP. The gap between LFF and scenario II with fixed update interval is due

to both higher uplink and downlink throughput of LFF. LFF provides higher uplink throughput

thanks to transmission of lower feedback compared to fixed update interval scheme. Also, in

scenario II, the UEs after 10 ms update the AP with one bit per subcarrier, and the AP does not

know the SINR value of each subcarrier to allocate them efficiently to the UEs.

C. LF Schemes Comparison

A comparison between the FF, one-bit, LCF and LFF schemes in case of transmitted overhead

is given in Table III. It is assumed that the SINR on each subcarrier can be fedback to the AP

using B bits, and M = [tu,opt/tfr]. Note that for M ≥ (B + 1), the overhead per frame of the

LFF scheme is lower than the one-bit feedback technique. Also, for M ≥ K/2, LFF scheme

produces lower overhead per frame in comparison to LCF. For N = 5, B = 10, tfr = 1.6 ms and

Rreq = 5 Mbps the overhead per frame versus different number of subcarriers are illustrated in

Fig. 9. The rest of parameters are the same as given in Table I and Table II. As can be observed

from Fig. 9, the FF scheme generates huge amount of feedback overhead especially for high

number of subcarriers. The practical one-bit feedback reduces the overhead by a factor of B. As

can be seen, the LCF always falls below the one-bit feedback method. The gap between LCF

and one-bit feedback becomes remarkable for higher number of subcarriers. The overhead results

of the LFF have been also presented for stationary UEs and UEs with low and normal speed.

Clearly, the LFF generates the lower feedback overhead per frame as the UE’s velocity tends

to zero. The expected sum-throughput of different feedback schemes with the same parameters

as mentioned above are summarized in Table. IV. As we expected, the LFF outperforms the

other schemes when the UEs are stationary. However, the sum-throughput of the LCF method

DRAFT July 14, 2018

25

TABLE III: Comparison of feedback schemes in case of overhead

Scheme Full feedback One-bit feedback Proposed LCF Proposed LFF

Overhead B(K/2− 1) bpf (K/2− 1) bpf B bpf B(K/2− 1)/M bpf

TABLE IV: Comparison of feedback schemes in case of expected sum-throughput, N = 5, Rreq = 5 Mbps and wu = wd = 1.

Scheme Full feedback One-bit feedback Proposed LCF Proposed LFF

Expected sum-throughput (v = 0 m/s) 6.67 Mbps 7.64 Mbps 8.33 Mbps 8.35 Mbps

Expected sum-throughput (v = 1 m/s) 6.67 Mbps 7.47 Mbps 8.33 Mbps 8.08 Mbps

Number of subcarriers128 256 512 1024 2048

Ove

rhea

d pe

r fr

ame,

[bi

ts]

10-1

100

101

102

103

104

FFOne-bit feedbackLCFLFF, v=1 m/sLFF, v=0.5 m/sLFF, v:0

Fig. 9: Transmitted overhead versus different number of subcarriers.

is higher for mobile UEs.

VI. CONCLUSION AND FUTURE WORKS

Two methods for reducing feedback cost were proposed in this paper: i) the limited-content

feedback (LCF) scheme, and ii) the limited-frequency feedback (LFF) method. The former is

based on reducing the content of feedback information by only sending the SINR of the first

subcarrier and estimating the SINR of other subcarriers at the AP. The latter is based on the

less frequent transmission of feedback information. The optimal update interval was derived,

which results in maximum expected sum-throughput of uplink and downlink. The Monte-Carlo

simulations confirmed the accuracy of analytical results. The effect of different parameters on

July 14, 2018 DRAFT

26

optimum update interval was studied. It was also shown that the proposed LCF and LFF schemes

provide better sum-throughput while transmitting lower amount of feedback compared to the

practical one-bit feedback method. The combination of the LCF with the update interval is the

topic of our future study.

APPENDIX

A. Proof of (25)

According to the RWP mobility model, the UE is initially located at P0 with the distance

r0 from cell center. The scheduler at the AP is supposed to allocate the resources to the UEs

as much as they require. Thus, the achievable data throughput of the UE at t = 0 is equal to

the requested data rate i.e., R(0) = Rreq. Hence, kreq can be obtained by solving the following

equation:

Rreq =Bd,n

K

kreq∑k=1

log2

Ge−4πkBd,nKw0

(h2 + r20)m+3

=Bd,n

K

kreq∑k=1

log2

(G

(h2 + r20)m+3

)+Bd,n

K

kreq∑k=1

log2

(e−4πkBd,nKw0

)

=kreqBd,n

Klog2

(G

(h2 + r20)m+3

)− 4π

w0

(Bd,n

K

)2

log2e

kreq∑k=1

k

=kreqBd,n

Klog2

(G

(h2 + r20)m+3

)− 2π

w0

(Bd,n

K

)2

(log2e)kreq(kreq + 1)

⇒ k2req+

1−log2

(G

(h2+r20)m+3

)2πBd,n

Kw0log2e

kreq+Rreq

2πw0

(Bd,n

K

)2log2e

=0.

(30)

The above equation is a quadratic equation and it has two roots where the acceptable kreq can

be obtained as follows:

kreq =

(log2

(G

(h2+r20)m+3

)2πBd,nKw0

log2e−1

)−

√√√√(1− log2

(G

(h2+r20)m+3

)2πBd,nKw0

log2e

)2

− 4Rreq

2πw0

(Bd,nK

)2log2e

2.

(31)

If Rreq � w0

8πlog2

(G

(h2+r20)m+3

), the approximate number of required subcarriers is kreq

∼=KRreq

Bd,n log2

(G

(h2+r20)m+3

) . With the parameters given in Table I, the constraint on the requested data

rate is Rreq << 350 Mbps.

DRAFT July 14, 2018

27

B. Proof of Proposition

The optimal solution of the OP given in (18) can be obtained by finding the roots of its

derivation that is ∂Er0,θ[T ]

∂tu= wu

∂Er0,θ[Ru]

∂tu+ wd

∂Er0,θ[Rd]

∂tu= 0. The expectation value of the

average downlink throughput is Er0,θ[Rd] =∫∫

r0,θRdfR0(r0)fΘ(θ)dθdr0 and its derivation is

equal to ∂Er0,θ[Rd]

∂tu= ∂

∂tu

∫∫r0,θ

RdfR0(r0)fΘ(θ)dθdr0. Since the function inside the integral

is derivative on the range (0, 2rc/v), the derivation operator can go inside the integral as∫∫r0,θ

∂Rd

∂tufR0(r0)fΘ(θ)dθdr0 [48], and this is the expectation value of the derivation of the

average downlink throughput, i.e., Er0,θ[∂Rd

∂tu]. Thus, we can conclude that ∂Er0,θ[Rd]

∂tu=Er0,θ[

∂Rd

∂tu].

Using the same methodology for uplink throughput we have ∂Er0,θ[Ru]

∂tu= Er0,θ[∂Ru

∂tu]. Then, the

derivation of (17) can be expressed as:

Er0,θ[∂T∂tu

]= wuEr0,θ

[∂Ru

∂tu

]+ wdEr0,θ

[∂Rd

∂tu

]. (32)

Hence, the root of Er0,θ[ ∂T∂tu ] = 0 will be the same as the root of ∂Er0,θ[T ]

∂tu= 0.

Using the Leibniz integral rule the derivation of (23) can be obtained as:

∂Ru

∂tu=−2(m+ 3)TuBu,n

Nt2u

(1−2tfb

tu

)(tu

2(m+ 3)log2

(Gu

(r2(tu) +h2)m+3

)+r0 cos θ

2vlog2

(r2(tu)+h2

)− (h2+r2

0 sin2θ)12

v ln(2)tan−1

(vtu − r0 cos θ

(h2+r20 sin2θ)

12

)− (h2+r2

0 sin2θ)12

v ln(2)tan−1

(r0 cos θ

(h2+r20 sin2θ)

12

)

−r0 cos θ

2vlog2

(r2

0 +h2)

+tu

ln(2)

)+TuBu,n

Ntu

(1−tfb

tu

)log2

(Gu

(r2(tu) +h2)m+3

)(33)

Using the sum of inverse tangents formula, tan−1(a) + tan−1(b) = tan−1(a+b1−ab

), (33) can be

further simplified as:

∂Ru

∂tu=−2(m+ 3)TuBu,n

Nt2u

(1−2tfb

tu

)(r0 cos θ

2vlog2

(r2(tu)+h2

r20 +h2

)

−(h2+r20 sin2θ)

12

v ln(2)tan−1

vtu

(h2+r20 sin2θ)

12

1− r0 cos θ(vtu− r0 cos θ)

h2 + r20 sin2θ

+tu

ln(2)

+TuBu,ntfbNtu

log2

(Gu

(r2(tu)+h2)m+3

).

(34)

This is the exact derivation of the average uplink achievable throughput respect to tu, however,

July 14, 2018 DRAFT

28

for vtu � h, this equation can be further simplified. Substituting r(tu) = (r20+v2t2u−2r0vtu cos θ+

h2)1/2 in logarithm term, ignoring the small terms and using the approximation ln(1 + x) ∼= x

for small values of x, we arrive log2

(1+v2t2u−2r0vtucosθ

r20+h2

)∼= log2

(1−2r0vtucosθ

r20+h2

)∼= −2r0vtucosθ

ln(2)(r20+h2).

Considering the rule of small-angle approximation for inverse tangent, it can also be approximated

by its first two terms of Taylor series as tan−1(x) ∼= x−x3/3 for small x. Noting that tfb � tu,

the approximate derivation is given as follows:

∂Ru

∂tu∼=−2(m+ 3)TuBu,n

ln(2)Nt2u

(1−2tfb

tu

)((vtu)3(h2+ r2

0 sin2θ)2

3v(h2 + r20)3

+ tu

−r20 cos2θtur2

0 +h2− tu(h2+r2

0 sin2θ)

h2 + r20

)+TuBu,ntfbNt2u

log2

(Gu

(r20 +h2)m+3

)= −2(m+ 3)TuBu,nv

2(h2 + r20 sin2θ)2tu

3N ln(2)(h2 + r20)3

+TuBu,ntfbNt2u

log2

(Gu

(r20 + h2)m+3

) (35)

Using the Leibniz integral rule to calculate the derivation of average downlink throughput, and

the sum of inverse tangents formula to simplify it, the derivation of average downlink throughput

is given as:

∂Rd

∂tu=−2(m+ 3)kreqBd,n

Kt2u

(r0cosθ

2vlog2

(r2(tu) + h2

r20 + h2

)+

tuln(2)

−(h2+r20 sin2θ)

12

v ln(2)tan−1

vtu

(h2 + r20 sin2 θ)

12

1− r0 cos θ(vtu−r0 cos θ)

(h2 + r20 sin2 θ)

.

(36)

This is the exact derivation of average downlink achievable throughput respect to tu, however,

using the approximation rules for vtu � h, the well-approximated derivation is given as follows:

∂Rd

∂tu∼=−2(m+ 3)kreqBd,n

Kt2u

(− r2

0 cos2 θtuln(2)(r2

0 + h2)− tu(h2 + r2

0 sin2 θ)

ln(2)(h2 + r20)

+(vtu)3(h2 + r2

0 sin2 θ)2

3v ln(2)(h2 + r20)3

+tu

ln(2)

)=−2(m+ 3)kreqBd,nv

2tu(h2 + r20 sin2 θ)2

3K ln(2)(h2 + r20)3

.

(37)

The exact optimum time, tu,opt, can be obtained numerically by solving (26) after substituting

∂Rd

∂tuand ∂Ru

∂tugiven in (33) and (36). However, we can approximately obtain a closed form

for optimum update interval denoted as tu,opt by using (35) and (37). Taking into account that

vtu � h the closed solution form for optimum update interval is given as:

DRAFT July 14, 2018

29

tu,opt∼=

( 3ln(2)2(m+3)

wutfbTuBu,nC1

wdv2NRreq + C2wuv2TuBu,n

)13

,

where

C1 =

Er0[log2

(Gu

(r20 + h2)m+3

)]Er0[log2

(G

(r20 + h2)m+3

)]Er0,θ

[(h2 + r2

0 sin2 θ)2

(h2 + r20)3

] , C2 = Er0[log2

(G

(r20 + h2)m+3

)].

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